These concepts are revisited in the rotational plane with changes to names but a similarity in concept.
Rotational vs. Revolving
Rotation: Movement around an internal axis (e.g., a bicycle tire rotating around its axis).
Revolution: Movement around an external axis (e.g., a red spot on the wheel revolving about the axis).
Earth’s Rotational Motion
The Earth revolves around the sun every 365 \frac{1}{4} days.
The Earth rotates around its axis every 24 hours.
Period (T)
Definition: Time for one revolution (or the time to go around once).
Units: Seconds, minutes, hours, or any unit of time.
Frequency
Definition: How often an object travels around a circle.
Units: Revolutions per minute (rpm) or revolutions per second (rps).
Relationship between Frequency and Period:
Period vs. Frequency - Example
If it takes 5 seconds for a dog to go around a merry-go-round once (the Period), we can determine the frequency.
The Radian
Definition: The angle for which the length of a circular arc is equal to the radius of the circle.
Units for measuring angles: Degree (°), revolution (rev), and radian (rad).
Radian is the most convenient unit for angle measurements in scientific calculations.
Arc Length Formula: S = r\theta
Describing Angular Motion
Angles are indicated around the circumference of the circle in both degrees and radians.
Radians are the preferred unit for Physics.
Circumference Formula: C = 2\pi r
Conversion: 1 revolution = 360° = 2\pi radians
Rotational vs. Tangential Velocity
Tangential velocity (v): Describes the motion of an object along the edge of a circle; direction is always along the tangent to that point.
V is dependent on the point's location relative to the axis of rotation.
Rotational velocity (\omega): Describes the motion of a rotating body.
The entire body rotates at the same \omega.
Rotational Velocity
Definition: The number of rotations per unit of time.
Formula: \omega = \frac{2\pi}{T}
Units: radians per second (rad/sec)
Also referred to as angular velocity.
Tangential Speed
Definition: The speed of an object moving along a circular path.
Formula: v = \frac{2\pi r}{T} = \omega r
Units: m/s (meters per second)
Rotational Inertia
An object rotating about an axis tends to remain rotating about the same axis at the same rotational speed unless interfered with by an external influence.
Definition: The property of an object to resist changes in its rotational state of motion (symbol I).
Also known as the Moment of Inertia.
Factors Affecting Rotational Inertia
Mass of the object.
Distribution of mass around the axis of rotation.
The greater the distance between an object’s mass concentration and the axis, the greater the rotational inertia.
Implications of Rotational Inertia
The greater the rotational inertia, the harder it is to change its rotational state.
Example: A tightrope walker carries a long pole with high rotational inertia for stability.
Rotational Inertia and Axis of Rotation
Rotational inertia depends on the axis around which it rotates.
Easier to rotate a pencil around an axis passing through it.
Harder to rotate it around a vertical axis passing through the center.
Hardest to rotate it around a vertical axis passing through the end.
Center of Gravity (CG)
Definition: The average position of an object's weight distribution.
For simple, uniform objects, the center of gravity is located at the geometric center.
The center of gravity can be located outside of an object.
Locating the Center of Gravity
An object hangs with the center of gravity below the point of suspension.
An object will balance if pivoted exactly above or below its center of gravity.
Balance on a pivot is stable if CG is below the pivot.
Human Center of Gravity
Standing upright, your CG is roughly in the center of your body (at about 55% of your height).
Location of your CG will shift when you bend your torso, move your arms and legs, etc.
Torque
Definition: The tendency of a force to cause rotation.
Depends upon three factors:
Magnitude of the force
The direction in which it acts
The point at which it is applied on the object
Torque Equation
Formula: Torque = radius × force
The radius depends upon where the force is applied and the direction in which it acts.
The force needs to be applied perpendicular to the radius.
Unit for torque is Newton-meter (N*m).
Lever Arm Examples
Lever arm is less than the length of the handle because of the direction of force.
Lever arm is equal to the length of the handle.
Lever arm is longer than the length of the handle.
Centripetal Force
Definition: Any force directed toward a fixed center.
Centripetal means “center-seeking” or “toward the center.”
Example: Whirling a tin can at the end of a string; you pull the string toward the center to keep the can moving in a circle.
Centripetal Force in Circular Motion
For an object moving in a circle, there is an inward force acting upon it in order to cause its inward acceleration.
For objects moving in circular motion, there is a net force acting towards the center which causes the object to seek the center.
Centripetal Force - Example
When a car rounds a curve, the centripetal force prevents it from skidding off the road.
If the road is wet, or if the car is going too fast, the centripetal force is insufficient to prevent skidding off the road.
Angular Momentum and Its Conservation
Systems that can change their rotational inertia through internal forces will also change their rate of rotation.
Formula: L = \omega * I
Angular Momentum - Example
If, by pulling the weights inward, the rotational inertia of a man reduces to half its value, by what factor would his angular velocity change?